At some point it might make sense to stop using SNPs, and start sequencing whole genomes. Collins suggests that the NHGRI's 1,000 genomes project, which aims to sequence the genomes of at least 1,000 people from all over the world, could go a long way towards finding hidden heritability, and many more genomes may become possible as the price of sequencing falls.

The genome is the genetic material that is present within a cell. The human genome is what roughly represents us as a species, but of course there is individual variation. Some people have blond hair, others brown, that's because pigments are encoded slightly differently in their genomes. Here you can find the genome of Craig Venter http://huref.jcvi.org one of the people who drove the human genome project.

A genome can even be different in a single human being, such individuals are called mosaic and we encounter them in clinical practice when a blood sample and a skin sample give discordant genetic information.

Then what is meant when we see a news story that announces that the genome of a certain species has been "sequenced"? Does this only mean that a genome of a particular individual of that species has been sequenced?

My understanding is that the genes are the same, but that 'gene' is a more general description of the molecular string, and that the alleles are what vary (specific segments of the gene). So there is (mostly) only one human genome but individual humans have different alleles.

Then what is meant when we see a news story that announces that the genome of a certain species has been "sequenced"? Does this only mean that a genome of a particular individual of that species has been sequenced?

It can mean both things, for humans it were individuals that were sequenced (which is of interest to correlate genotype with phenotypes). In a recent example the hookworm genome was sequenced, in that case worms were ground in liquid nitrogen and solubilized in buffer: http://www.nature.com/ng/journal/vaop/ncurrent/pdf/ng.2875.pdf The worms are probably inbred, so in that case one can take a population, because they don't vary much genetically anyway.

I've worked with an isogenic (inbred) nematode strain that was sequenced by a consortium, later we found out that over generations our lab strain (a clone) had accumulated many mutations and that made it impossible to answer a certain research question: we had to sequence it again. It's impossible to say that there is "a genome for a species", one has to take variation and mutation in account.